J. Chem. Thermodynamics 43 (2011) 51–57
Contents lists available at ScienceDirect
J. Chem. Thermodynamics
journal homepage: www.elsevier.com/locate/jct
Thermodynamic properties of strontium titanates: Sr2TiO4, Sr3Ti2O7, Sr4Ti3O10,
and SrTiO3
K.T. Jacob ⇑, G. Rajitha
Department of Materials Engineering, Indian Institute of Science, Bangalore 560 012, India
a r t i c l e
i n f o
Article history:
Received 29 April 2010
Received in revised form 29 July 2010
Accepted 5 August 2010
Available online 18 August 2010
Keywords:
Strontium titanates
Gibbs free energy
Enthalpy
Entropy
Thermodynamic properties
EMF technique
a b s t r a c t
The chemical potentials of SrO in two-phase fields (TiO2 + SrTiO3), (SrTiO3 + Sr4Ti3O10), (Sr4Ti3O10 + Sr3Ti2O7), and (Sr3Ti2O7 + Sr2TiO4) of the pseudo-binary system (SrO + TiO2) have been measured in the temperature range (900 to 1250) K, relative to pure SrO as the reference state, using solid-state galvanic cells
incorporating single crystal SrF2 as the electrolyte. The cells were operated under pure oxygen at ambient
pressure. The standard Gibbs free energies of formation of strontium titanates, SrTiO3, Sr4Ti3O10, Sr3Ti2O7,
and Sr2TiO4, from their component binary oxides were derived from the reversible electromotive force
(EMF) of the cells. For the formation of the four compounds from their component oxides TiO2 with rutile
structure and SrO, the standard Gibbs free energy changes are given by:
1
DGðoxÞ ðSrTiO3 Þ 89=ðJ mol Þ ¼ 121878 þ 3:881ðT=KÞ;
DGðoxÞ ðSr4 Ti3 O10 Þ
1
284=ðJ mol Þ ¼ 409197 þ 14:749ðT=KÞ;
1
DGðoxÞ ðSr3 Ti2 O7 Þ 190=ðJ mol Þ ¼ 285827 þ 10:022ðT=KÞ;
DGðoxÞ ðSr2 TiO4 Þ
1
110=ðJ mol Þ ¼ 159385 þ 3:770ðT=KÞ:
The reference state for solid TiO2 is the rutile form. The results of this study are in good agreement with
Gibbs free energy of formation data reported in the literature for SrTiO3, but differ significantly with data
for Sr4Ti3O10. For Sr3Ti2O7 and Sr2TiO4 experimental measurements are not available in the literature for
direct comparison with the results obtained in this study.
Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction
High dielectric constant of (330) and low dielectric loss of the
order of 103 is characteristic of SrTiO3 at room temperature and
50 MHz frequency [1]. Hence, it can be used as a dielectric material
in high performance metal–insulator–metal (MIM) capacitors for
analog applications [2]. It shows relatively stable dielectric properties (effective dielectric constant of 5 104) and low dielectric
loss (0.005) over the temperature range (243 to 373) K and frequency range (1 to 100) kHz when excess of TiO2 is present or
Nb2O5 is used as an additive [3]. SrTiO3 shows semi-conduction
characteristics when it is doped with Al and Nb [4–6]. Domen
et al. [7] have reported the use of SrTiO3 along with 1.5 wt% NiO
as a photo-catalyst in the evolution of H2 and O2 from distilled
water. Polycrystalline samples of Sr2TiO4 and Sr3Ti2O7 have reasonably high dielectric constants (38 and 52, respectively) and low
⇑ Corresponding author. Tel.: +91 80 22932494; fax: +91 80 23600472.
E-mail address: [email protected] (K.T. Jacob).
0021-9614/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jct.2010.08.011
dielectric loss of the order of 104 at room temperature and
1.5 MHz frequency [8].
McCarthy et al. [9] have studied the system SrTiO at
T = 1648 K and reported the presence of four phases SrTiO3,
Sr4Ti3O10, Sr3Ti2O7, and Sr2TiO4. They were able to prepare Sr2TiO4,
Sr3Ti2O7, and SrTiO3 by holding appropriate mixtures of SrCO3 and
TiO2 at T = 1648 K. However, heating to T = 1748 K was required to
get the single phase of Sr4Ti3O10. Tilley [10] was able to obtain only
three compounds Sr2TiO4, Sr3Ti2O7, and SrTiO3 from SrCO3 and
TiO2 in the temperature range (1513 to 1673) K. Wei et al. [11]
have studied the ternary system SrOTiO2B2O3 below T =
1300 K and reported only three phases Sr2TiO4, Sr3Ti2O7, and
SrTiO3 along the binary SrOTiO2 in air. In an earlier study,
Ruddlesden and Popper [12,13] have obtained Sr3Ti2O7 and Sr2TiO4
by reacting SrCO3 and TiO2 in the appropriate ratios at T = 1673 K.
SrTiO3 is a cubic perovskite with space group Pm3m and lattice
parameter a = 0.3905 (8) nm are given in JCPDS (File No. 35-734).
Sr4Ti3O10 has a tetragonal structure with the space group I4/
mmm and lattice parameters a = 0.3903 nm, c = 2.814 nm [9].
Elcombe et al. [14] have refined lattice parameters of tetragonal
Sr3Ti2O7 having double perovskite structure with the space group
52
K.T. Jacob, G. Rajitha / J. Chem. Thermodynamics 43 (2011) 51–57
I4/mmm as a = 0.39026 nm and c = 2.03716 nm using neutron diffraction. Sr2TiO4 has a quasi-two-dimensional K2NiF4 structure
(Ruddlesden–Popper type) [13] with the space group I4/mmm
and lattice parameters a = 0.388 nm, c = 1.26 nm as reported in
JCPDS (File No. 39-1471).
Significant amount of thermodynamic data are available in the
literature for SrTiO3 and very limited data for Sr2TiO4 and
Sr4Ti3O10. Todd and Lorenson [15] measured heat capacity of
SrTiO3 in the temperature range (51 to 300) K and derived the entropy of SrTiO3 at T = 298.16 K as (108.78 ± 0.84) J K1 mol1.
They found no anomaly in the variation of heat capacity with temperature. However, heat capacity measurements of Salje et al. [16]
in the temperature range (85 to 120) K detected a small anomaly at
T = 105.65 K corresponding to the cubic-tetragonal (I4/mcm) phase
transition. Gallardo et al. [17] also confirmed the structural phase
transition at T = 105 K. Two different techniques were used to measure heat capacity; quasi-static conduction calorimetry in the temperature range (85 to 120) K and AC calorimetry in the range (4 to
120) K [17]. Recently, Duran et al. [18] have studied low-temperature heat capacities in the temperature range (2 to 300) K by a
relaxation method. They found minor anomalies at T = (37 and
105) K in the specific heat of SrTiO3. High-temperature heat contents (HT H298.15) of SrTiO3 and Sr2TiO4 in the temperature range
(384 to 1831) K were measured by Coughlin and Orr [19]. Ligny
and Richet [20] have also determined high-temperature heat content (HT H298.15) of cubic SrTiO3 in the temperature range (300 to
1800) K using drop calorimetry. Xu et al. [21] have determined the
enthalpy of formation of SrTiO3 at T = 298 K from component binary oxides as (117.1 ± 2.1) kJ mol1 by drop solution calorimetry
in sodium molybdate at T = 974 K. Determined by Panfilov and
Feodosev [22] using bomb calorimeter was the enthalpy change
for the reaction, SrCO3 (solid) + TiO2 (solid) ? SrTiO3 (solid) + CO2
(gas). The standard enthalpy of formation of SrTiO3 from elements
at T = 298.15 K derived from this measurement was (1668.99 ±
9.2) kJ mol1. Combining with data for SrO and TiO2 from NIST-JANAF tables [23], the enthalpy of formation from component binary
oxides is obtained as (132.21 ± 9.5) kJ mol1. Taylor and Schmalzried [24] measured the Gibbs free energy of formation for
SrTiO3 at T = 833 K using a solid-state cell incorporated with SrF2
as solid electrolyte. They have also estimated the Gibbs free energy
of formation for Sr4Ti3O10 at T = 813 K assuming it to be equilibrium with SrTiO3. Thermodynamic data for SrTiO3, Sr4Ti3O10, and
Sr2TiO4 are given in the compilation of Knacke et al. [25], some
based on measurement and others on estimation. There is no thermodynamic information on Sr3Ti2O7.
2. Experimental methods
2.1. Materials
The starting materials were SrCO3 and TiO2 of mass fraction
purity >0.9999. The rutile form of TiO2 was dried at T = 1073 K
and SrCO3 at T = 573 K before use. Strontium titanates, SrTiO3,
Sr4Ti3O10, Sr3Ti2O7, and Sr2TiO4, were prepared by heating mixtures of SrO and TiO2 in the appropriate molar ratio contained in
platinum crucibles at T = 1753 K for 160 ks in air with three intermediate grindings and compactions. A reactive form of SrO, obtained by decomposition of SrCO3 in vacuum (p 1 Pa) at
T = 1500 K, was used in the preparation of ternary oxides. The mixtures of component oxides were pelletized at p = 150 MPa using a
steel die before heating. Completion of the reaction was confirmed
by powder X-ray diffraction (XRD) analysis of the product: SrTiO3
with cell parameter a = 0.3904 nm; Sr4Ti3O10 with cell parameters
a = 0.3905 nm, c = 2.8146 nm; Sr3Ti2O7 with cell parameters
a = 0.3902 nm and c = 2.0373 nm; Sr2TiO4 with cell parameters
a = 0.3884 nm and c = 1.2605 nm. Transparent single crystals of
SrF2 for use as solid electrolytes were obtained in the form of discs,
1.5 cm in diameter and 0.3 cm thick. The high-purity oxygen gas,
used to fix the oxygen potential over the electrodes of the solidstate cell, was passed through sodium hydroxide to remove traces
of CO2 and then dried by passing through columns containing
anhydrous magnesium perchlorate and phosphorus pentoxide.
2.2. Apparatus and procedure
The reversible EMF of four solid-state cells,
Au; O2 ; SrO þ SrF2 =SrF2 =SrF2 þ SrTiO3 þ TiO2 ; O2 ; Au;
Au; O2 ; SrO þ SrF2 =SrF2 =SrF2 þ Sr4 Ti3 O10 þ SrTiO3 ; O2 ; Au;
ðIÞ
ðIIÞ
Au; O2 ; SrO þ SrF2 =SrF2 =SrF2 þ Sr3 Ti2 O7 þ Sr4 Ti3 O10 ; O2 ; Au; ðIIIÞ
Au; O2 ; SrO þ SrF2 =SrF2 =SrF2 þ Sr2 TiO4 þ Sr3 Ti2 O7 ; O2 ; Au
ðIVÞ
were measured as a function of temperature in the range (900 to
1250) K. The cells are written such that the right-hand electrodes
are positive. The reference electrode consisted of an intimate equimolar mixture of SrO and SrF2. The mixture was pelletized at
p = 100 MPa using a steel die and sintered in a stream of pre-purified oxygen at T = 1300 K. The measuring electrodes consisting of
three phase mixtures of SrF2 + SrTiO3 + TiO2, SrF2 + Sr4Ti3O10 + SrTiO3, SrF2 + Sr3Ti2O7 + Sr4Ti3O10, and SrF2 + Sr2TiO4 + Sr3Ti2O7 in
the molar ratio 0.5:1:1 were also pelletized and sintered under
identical conditions. Minor variation in the mixing ratio of the constituents forming the electrode did not affect the cell EMF. The presence of SrF2 in the electrode pellets was found necessary to generate
fluorine chemical potentials at the electrodes. Since fluorine ions
are the mobile species in the SrF2 solid electrolyte, the cells respond
to the difference in fluorine chemical potential at the two electrodes. The apparatus used in this study for EMF measurements
was identical to that used recently in the study of calcium titanates
[26] and similar to those described elsewhere [27,28]. Only some
additional details and critical points required for evaluation of the
experiments are given here. The electrode pellets were spring
loaded on either side of transparent single crystal SrF2 electrolyte
with a gold mesh sandwiched between each electrode pellet and
the electrolyte. Gold electrical leads were spot-welded to the mesh.
The presence of the gold catalyst at the electrolyte–electrode interface was found necessary to obtain reproducible EMF. Without the
catalyst, the EMF of the cell was found to be lower than the equilibrium value, especially at lower temperatures, and the response was
significantly slower. The pellets were held together under pressure
by a system of alumina tubes and rods. Au foils were placed to prevent physical contact between the electrode pellets and alumina
rods and tubes used for holding the pellets under pressure. The cell
was enclosed in an outer impervious alumina tube, closed at both
ends with brass caps, which had provision for gas inlet and outlet,
and electrode and thermocouple leads. The alumina tube was suspended in a vertical resistance furnace. The cell was situated in
the even-temperature zone (±1 K) of the furnace. A Faraday cage
made of stainless steel foil was placed between the alumina tube
and the furnace. The foil was earthed to minimize induced EMF
on cell leads from furnace winding.
After assembling the cell and raising its temperature to
T = 573 K under flowing oxygen gas, the outer alumina tube enclosing the cell was evacuated and then refilled with oxygen gas. The
procedure was repeated three times to remove moisture that desorbed from the ceramic tubes. The cell was then operated under
pre-purified oxygen gas flowing at a rate of (3) ml s1. The same
gas flowed over both electrodes. The cell EMFs were measured as a
function of temperature using a high-impedance (>1012 X) digital
voltmeter with a sensitivity of ±0.01 mV. The EMF of the cell was
independent (±0.2 mV) of the flow rate of the gas in the range
53
K.T. Jacob, G. Rajitha / J. Chem. Thermodynamics 43 (2011) 51–57
(1.4 to 4.7) ml s1 and was reproducible after temperature cycling. The temperature of the cell was measured with a Pt/Pt–
13%Rh thermocouple checked against the melting temperature of
gold.
The EMF of the cells became steady in (6 to 24) ks after the
attainment of thermal equilibrium, depending on the temperature
of the cell. The reversibility of each cell at different temperatures
was checked by microcoulometric titration (5 lA for 180 s) in
both directions using an external potential source. In each case,
the cell EMF returned to its original value before the titration within 2 ks. This demonstrated that the chemical potential of fluorine at
the electrodes returned to the same value after essentially infinitesimal displacements from equilibrium to both lower and higher
values. To check for the presence of any thermal gradient across
the cell and other stray contributions to the EMF, the open circuit
potential of a symmetric cell with identical electrodes
Au; O2 ; SrO þ SrF2 =SrF2 =SrF2 þ SrO; O2 ; Au
ðVÞ
was measured as a function of temperature. The EMF was found to
lie in a narrow range, (±0.2 mV), without any systematic trends.
Thermodynamic data [23] indicate that SrF2 is stable in contact with
SrO and TiO2 in dry oxygen. At the end of each experiment, the cell
was cooled and the electrodes were examined by optical and scanning electron microscopy and XRD. No change in the phase composition of the electrodes during the electrochemical measurement
was detected. The composition of different phases present in the
electrodes was checked by energy dispersive analysis of X-rays
(EDX). There was no significant solid solubility either between the
oxides or between oxides and SrF2 present at the electrodes. There
was no evidence of reaction between SrF2 single crystal and the
electrode pellets at temperatures up to T = 1250 K. The most important factor for successful operation of the EMF cell was the removal
of trace amounts of CO2 and H2O from the gas phase. Reaction of
SrF2 electrolyte with moisture resulted in an opaque coating of
SrO on the electrolyte surface and deterioration in the performance
of the cell.
FIGURE 1. Temperature dependence of the reversible EMF of solid-state cells: —h—
(red online), cell (I); —N— (blue online), cell (II); —s— (green online), cell (III); and
—j— (black online), cell (IV). (For interpretation of the references in colour in this
figure legend, the reader is referred to the web version of this article.)
TABLE 1
The values of the coefficients a, b and EX at T = 1200 K along with uncertainty
estimates obtained from the regression analysis of EMF of each cell represented by
the expression EX/(mV) = a + b(T/K).
Cell
E/mV at T = 1200 K
a
b
I
II
III
IV
607.47 ± 0.46
206.43 ± 0.51
199.03 ± 0.22
186.15 ± 0.44
631.60 ± 0.6
225.75 ± 0.65
202.56 ± 0.28
170.72 ± 0.57
0.02011 ± 0.00056
0.0161 ± 0.00061
0.00294 ± 0.00026
0.01286 ± 0.00052
3. Results and discussion
The reversible EMFs of cells (I, II, III, IV) are displayed in figure 1 as a function of
temperature. At T 900 K, the cell EMFs were monitored for periods up to 90 ks.
The EMF remained constant during this entire period. Within experimental uncertainty, the EMFs can be expressed as linear functions of temperature as
EX =ðmVÞ ¼ a þ bðT=KÞ;
ð1Þ
where the subscript X denotes the cell number. The values of a, b and E at T = 1200 K
along with the corresponding uncertainty estimates obtained from linear regression
analysis of the EMF of each cell are presented in table 1. The quoted uncertainties
correspond to twice the standard error estimate (2r) obtained from the regression
analysis.
The electrochemical reaction at the measuring electrode on the right-hand side
of cell I is
SrF2 ðsolidÞ þ 1=2O2 ðgasÞ þ TiO2 ðsolidÞ þ 2e ! SrTiO3 ðsolidÞ þ 2F :
ð2Þ
At the reference electrode on the left-hand side of the cell, the electrochemical reaction is
SrO ðsolidÞ þ 2F ! SrF2 ðsolidÞ þ 1=2O2 ðgasÞ þ 2e :
ð3Þ
Since oxygen partial pressure is the same over both the electrodes, the net cell reaction is
SrO ðsolidÞ þ TiO2 ðsolidÞ ! SrTiO3 ðsolidÞ:
ð4Þ
Throughout this paper the reference state for solid TiO2 is the rutile form. The cell
reaction can also be formulated as the transfer of one mole of SrO at unit activity
(standard state) on the left-hand side of the cell to SrO at reduced activity corresponding to a invariant mixture of phases at constant temperature on the right-hand
side of the cell. Viewed from this perspective, the cell EMF is related to the difference
in the chemical potential of SrO at the two electrodes: lSrO lSrO ¼ 2FE. SrF2 is a
fluorine ion conductor, with ionic transport number greater than 0.99, at the temperatures and fluorine chemical potentials encountered in this study [29]. Since there
was no significant solid solubility between SrF2, SrO, TiO2, SrTiO3, Sr4Ti3O10, Sr3Ti2O7,
and Sr2TiO4 at the temperatures covered in this study, the compounds are present at
unit activity at the electrodes. The standard Gibbs free energy change associated
with the net cell reaction is directly related to the EMF of cell by the Nernst equation;
the standard Gibbs free energy change for reaction , DGrð4Þ , is related to the EMF of
cell I
1
DGrð4Þ 89=ðJ mol Þ ¼ 2FEI ¼ 121878 þ 3:8806ðT=KÞ;
ð5Þ
where F is the Faraday constant. The temperature-dependent term on the right-hand
side of equation (5) gives a ‘‘second-law” entropy of formation of SrTiO3 from oxides
obtained in this study (3.8806 ± 0.107) J mol1 K1 at Tav = 1075 K. The small
negative entropy of formation makes the Gibbs free energy of formation marginally
less negative with increasing temperature. The temperature independent term on
the right-hand side of equation (5) gives a ‘‘second-law” enthalpy of formation of
SrTiO3 from its component binary oxides (121.88 ± 0.115) kJ mol1 at a mean
1
temperature of 1075 K. At T = 833 K, the value of DGrð4Þ 89=ðJ mol Þ ¼ 118646
obtained from this study compares with the value of (117570 ± 4184) J mol1 reported by Taylor and Schmalzried [24], who used an almost identical cell with polycrystalline SrF2 as the electrolyte. However, as shown in figure 2, the free energy
values obtained in this study are significantly higher, (15.9) kJ mol1, than those
given in the compilation of Knacke et al. [25]. The reason for this large difference
is identified in the subsequent discussion.
Since several low- and high-temperature heat capacity measurements [15–18]
are now available in the literature for SrTiO3, a ‘‘third-law” analysis of the results of
this study can be undertaken. The low-temperature heat capacity measurements
are compared in figure 3. The earlier results of Todd and Lorenson [15] in the temperature range (51 to 300) K and the recent results of Duran et al. [18] in the range
(2 to 300) K are in fair agreement, with the values of Duran et al. slightly higher.
Measurements of Salje et al. [16] in the temperature range (85 to 120) K and
Gallardo et al. [17] in the range (4 to 120) K give almost identical results, which
are significantly lower than those of Todd and Lorenson [15] and Duran et al.
[18]. The data of Salje et al. [16] and Gallardo et al. [17] extrapolated to
T = 298.15 K does not match with high-temperature heat capacity data. Heat
capacity anomalies associated with phase transitions detected by Salje et al. [16]
54
K.T. Jacob, G. Rajitha / J. Chem. Thermodynamics 43 (2011) 51–57
FIGURE 2. The standard Gibbs free energy of formation of SrTiO3 from its
component binary oxides as a function of temperature: —d— (red online), this
study; N (blue online), Taylor and Schmalzried [24]; and —j— (black online),
Knacke et al. [25]. (For interpretation of the references in colour in this figure
legend, the reader is referred to the web version of this article.)
FIGURE 3. The variation of the low-temperature heat capacity of SrTiO3 with
temperature: h (red online), Todd and Lorenson [15]; s (blue online), Salje et al.
[16] and Gallardo et al. [17]; + (green online), Duran et al. [18]; and — (black online),
selected data. (For interpretation of the references in colour in this figure legend,
the reader is referred to the web version of this article.)
and Duran et al. [18] are almost invisible in the scale used in figure 3. The heat
capacity reassessed in this study, based on closely agreeing results of Todd and
Lorenson [15] and Duran et al. [18] and consistent with data at higher temperatures,
is shown by the continuous curve in the figure. The standard entropy of SrTiO3
at T = 298.15 K obtained from the reassessed heat capacity data is (109.128 ±
0.7) J K1 mol1.
The high-temperature heat capacity values derived from experiments [19,20]
are compared in figure 4. The results are in reasonable agreement, and the selected
values fall between the two sets of data and join smoothly with the low-temperature heat capacity data. The high-temperature heat capacity can be represented by
the equation:
1
C P =ðJ K1 mol Þ ¼ 137:09676 þ 0:00323T 456:79197T 0:5 1195220T 2 :
ð6Þ
Based on the selected values of C P ðTÞ, values of HT H298.15 and ST S298.15 for
SrTiO3 are evaluated and presented in table 2. These values are used for ‘‘third-law”
analysis of the standard Gibbs free energy of formation of SrTiO3 measured in this
study using the relation,
DHfðoxÞ;298:15 ¼ DGfðoxÞ ðTÞ DðHT H298:15 Þ þ TfDSfðoxÞ;298:15 þ DðST S298:15 Þg;
ð7Þ
FIGURE 4. High-temperature heat capacity of SrTiO3 as a function of temperature:
j (blue online), Coughlin and Orr [19]; h (red online), Ligny and Richet [20];
and — (black online), selected data, equation (6). (For interpretation of the
references in colour in this figure legend, the reader is referred to the web version
of this article.)
where DGfðoxÞ ðTÞ ¼ DGrð4Þ ðTÞ is the standard Gibbs free energy of formation of SrTiO3
from its component binary oxides, and DðHT H298:15 Þ and DðST S298:15 Þ represents
P
P
the difference of ðHT H298:15 Þ and ðST S298:15 Þ between products and reactants,
each multiplied by the appropriate stoichiometric coefficient. For assessing the entropy of formation of SrTiO3 from binary oxides at 298.15 K, DSfðoxÞ;298:15 , and values
of DðHT H298:15 Þ and DðST S298:15 Þ, auxiliary data on SrO and TiO2 (rutile) are taken
from NIST-JANAF tables [23]. Using the selected standard entropy of SrTiO3 at
298.15, the standard entropy of formation from binary oxides DSfðoxÞ;298:15 ¼
1
3:314 0:863J K1 mol The results of the ‘‘third-law” analysis are displayed in
figure 5, where the value of enthalpy of formation calculated from values of free energy of formation at different temperatures is plotted against the temperature of the
free energy measurement. Considering the uncertainty in the data used for the analysis, the enthalpy of formation is essentially constant, DHfðoxÞ;298:15 ¼ 116:507
1
0:9 kJ mol . The uncertainty in entropy is the major contributor to the error in
the ‘‘third-law” enthalpy of formation. The small variation of 0.52 kJ mol1 in
DHfðoxÞ;298:15 with temperature suggests either a small temperature-dependent error
in EMF or an inaccuracy of 1.3JK1 mol1 in entropy of formation. This small
difference is also reflected in the enthalpy of formation from oxides at T = 298 K ob1
tained by ‘‘second-law” ðDHfðoxÞ;298:15 ¼ 118:032 0:25 kJ mol Þ and ‘‘third-law”
1
ðDHfðoxÞ;298:15 ¼ 116:507 0:9 kJ mol Þ techniques.
As shown in figure 5, the value of DHfðoxÞ;298:15 obtained from the ‘‘third-law”
analysis is in good agreement with the high-temperature solution calorimetric
determination of Xu et al. [21], but differs significantly from the bomb calorimetric
value reported by Panfilov and Feodosev [22]. Since the data compiled by Knacke
et al. [25] for SrTiO3 is based on the bomb calorimetric value of enthalpy of formation [22], the free energy of formation given in compilation is more negative than
that obtained in this study as shown in fig. 2. Thus the results of this study strongly
suggest revision of data for SrTiO3 given by Knacke et al. [25]. Complete data for
SrTiO3 based on the results of this study, assessed heat capacity and auxiliary data
for SrO and TiO2 from NIST-JANAF is provided in table 2. The table lists the enthalpy
and free energy of formation of SrTiO3 from elements at regular intervals of temperature. The entropy of SrTiO3 assessed from thermal data at is marginally higher by
(1.43)JK-1mol-1 than the value obtained in this study at Tav = 1075 K from EMF
measurement.
The EMF of cell II is related to the standard Gibbs free energy change for the
reaction
SrO ðsolidÞ þ 3SrTiO3 ðsolidÞ ! Sr4 Ti3 O10 ðsolidÞ;
1
DGrð8Þ 98=ðJ mol Þ ¼ 2FEII ¼ 43563 þ 3:107ðT=KÞ:
1
ð8Þ
ð9Þ
At T = 813 K the value of DGrð8Þ 98 ðJ mol Þ ¼ 41037 obtained in this study is
substantially different from the value of (90, 374 ± 4, 184) J mol1 reported by
Taylor and Schmalzried [24] using a cell similar to that employed in this study,
but with polycrystalline SrF2 electrolyte and Pt electrodes. The formation of Sr4PtO6
is possible when SrO at relatively high activity is in contact with Pt metal and O2 gas
[30,31]. Although the formation of strontium platinum oxides would be retarded
considerably in the presence of SrF2, the possibility of side reactions affecting the
EMF remains. Taylor and Schmalzried [24] were able to obtain stable EMF only at
T = 813 K. The use of Au electrodes in this study was to avoid this potential complication. Further, Taylor and Schmalzried [24] could not confirm the presence of
55
K.T. Jacob, G. Rajitha / J. Chem. Thermodynamics 43 (2011) 51–57
TABLE 2
Thermodynamic properties of solid SrTiO3.
T/K
C P =ðJ K1 mol
298.16
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
1600
1700
1800
99.292
98.413
108.079
113.503
117.066
119.653
121.663
123.302
124.687
125.889
126.956
127.919
128.801
129.616
130.378
131.095
131.775
1
Þ
(HT H298.15)/(kJ mol1)
S/(J K1 mol1)
DHf =ðkJ mol
0
0.182
10.561
21.661
33.200
45.043
57.112
69.363
81.764
94.294
106.937
119.682
132.519
145.440
158.440
171.514
184.658
109.128
109.736
139.528
164.276
185.304
203.554
219.667
234.095
247.160
259.101
270.101
280.302
289.815
298.729
307.119
315.045
322.557
1653.290
1653.286
1652.707
1651.763
1650.720
1649.680
1648.680
1648.402
1647.349
1654.334
1658.455
1658.011
1657.600
1657.237
1656.939
1793.637
1791.655
1
Þ
1
DGf =ðkJ mol
Þ
1568.303
1567.776
1539.345
1511.109
1483.077
1455.219
1427.507
1399.842
1372.284
1344.457
1316.163
1287.654
1259.181
1230.736
1202.314
1172.728
1136.260
DHf and DGf are the standard enthalpy and Gibbs free energy of formation of SrTiO3 from elements.
be evaluated as (358.207 ± 1.714) J mol1 K1. It would be interesting to measure
the enthalpy of formation and heat capacity of Sr4Ti3O10 to cross-check data obtained in this study.
A comparison of the standard Gibbs free energy of formation of Sr4Ti3O10 from
its component binary oxides obtained in this study with values reported in the literature is presented in figure 6. It reveals that the results of this study are far more
negative than the value suggested by Taylor and Schmalzried [24], and significantly
more positive than the data given in the compilation of Knacke et al. [25]. The average difference in free energy of formation of Sr4Ti3O10 between the values obtained
in this study and Knacke et al. [25] is 33.76 kJ mol1.
The EMF of cell III is related to the standard Gibbs free energy change for the
reaction
SrO ðsolidÞ þ 2Sr4 Ti3 O10 ðsolidÞ ! 3Sr3 Ti2 O7 ðsolidÞ;
1
DGrð12Þ 42 ðJ mol Þ ¼ 2FEIII ¼ 39088 þ 0:5673ðT=KÞ:
ð12Þ
ð13Þ
The standard Gibbs free energy of formation of Sr3Ti2O7 from its component
binary oxides can be obtained by combining equations (10) and (12). For the
reaction
3SrO ðsolidÞ þ 2TiO2 ðsolidÞ ! Sr3 Ti2 O7 ðsolidÞ;
1
DGrð14Þ 190 ðJ mol Þ ¼ 285827 þ 10:022ðT=KÞ:
FIGURE 5. Comparison of the results of ‘‘third-law” analysis of the measured free
energy of formation of SrTiO3 with calorimetric determinations of enthalpy of
formation: j (black online), Present study (‘‘third-law” analysis); (red online),
Xu et al. [21]; and — (green online), Panfilov and Feodosev [22]. (For interpretation
of the references in colour in this figure legend, the reader is referred to the web
version of this article.)
ð14Þ
ð15Þ
The ‘‘second-law” enthalpy of formation of Sr3Ti2O7 from its component binary oxides at an average temperature Tav = 1075 K is (285.827 ± 0.247) kJ mol1. Assuming
Sr4Ti3O10 at the working electrode of their cell; the formation of the compound was
assumed based on the quantities of SrCO3 and TiO2 used in the preparation of the
electrode by heating the mixture at T = 1973 K for 3.6 ks. According to the results
of this study discussed later, the only two oxide phases stable at this temperature
are SrTiO3 and Sr2TiO4. The higher chemical potential of SrO suggested by the measurements of Taylor and Schmalzried [24] on cell II is indicative of either incomplete
formation of Sr4Ti3O10 or formation of ternary oxides richer in SrO [24]. In view of
these uncertainties in the earlier measurement [24], the data obtained in this study
is considered to be more reliable. The standard Gibbs free energy of formation of
Sr4Ti3O10 from its component binary oxides can be obtained by combining equations
(4) and (8). For the reaction
4SrO ðsolidÞ þ 3TiO2 ðsolidÞ ! Sr4 Ti3 O10 ðsolidÞ;
1
DGrð10Þ 284=ðJ mol Þ ¼ 409197 þ 14:749ðT=KÞ:
ð10Þ
ð11Þ
The stability of the compound, relative to the binary oxides, decreases with temperature. The ‘‘second-law” enthalpy of formation of Sr4Ti3O10 from its component binary oxides at an average temperature of 1075 K is (409.197 ± 0.37) kJ mol1. If the
Neumann–Koop rule is used to estimate the heat capacity of Sr4Ti3O10, then its standard enthalpy of formation at T = 298.15 K from component binary oxides would
also be the same as that at high temperature. Combining with data for SrO and
TiO2 [23], the standard enthalpy of formation of Sr4Ti3O10 from elements at
T = 298.15 K is (5611.598 ± 13.769) kJ mol1. Similarly, from the ‘‘second-law” entropy change for reaction (10), the standard entropy of Sr4Ti3O10 at T = 298.15 K can
FIGURE 6. The standard Gibbs free energy of formation of Sr4Ti3O10 from its
component binary oxides: —d— (red online), this study; N (blue online), Taylor and
Schmalzried [24]; and —j— (black online), Knacke et al. [25]. (For interpretation of
the references in colour in this figure legend, the reader is referred to the web
version of this article.)
56
K.T. Jacob, G. Rajitha / J. Chem. Thermodynamics 43 (2011) 51–57
Figure 7 compares the results obtained in this study with values from the compilation of Knacke et al. [25]. The results of this study are more negative. The difference
between the two data sets increases with temperature. The mean difference between the two sets is 4.32 kJ mol1.
The ‘‘second-law” enthalpy of formation of Sr2TiO4 from its component binary
oxides at an average temperature Tav = 1075 K is (159.38 ± 0.137) kJ mol1. Combining the measured enthalpy increments of Sr2TiO4 [19] with data for SrO and TiO2
[23], the ‘‘second-law” enthalpy of formation at T = 298.15 K is obtained as
(152.376 ± 0.185) kJ mol1. The corresponding standard enthalpy of formation
of Sr2TiO4 from elements at T = 298.15 K is (2281.195 ± 6.78) kJ mol1. The ‘‘second-law” entropy change for reaction (18) is (3.77 ± 0.13) J mol1 K1 at Tav =
1075 K, and (6.182 ± 0.17) J mol1 K1 at T = 298.15 K. The corresponding standard entropy of Sr2TiO4 at T = 298.15 K can be evaluated as (167.518 ±
0.858) J mol1 K1. It would be useful to confirm this value by low-temperature
heat capacity measurements. The reasonably good agreement between the calorimetrically derived entropy and that obtained from EMF measurement for SrTiO3,
gives confidence in the value of entropy of Sr2TiO4 obtained in this study. A thermodynamic data table for Sr2TiO4, composed from derived values of enthalpy of formation and standard entropy of Sr2TiO4 at T = 298.15 K and high-temperature heat
content data available [19] in the literature, is displayed as table 3. Corresponding
data tables for Sr4Ti3O10 and Sr3Ti2O7 are not presented since there is no experimental data on heat capacity or heat content of these compounds; estimates of
the standard entropy and enthalpy of formation at T = 298.15 K given in this paper
are based on the Neumann–Koop rule.
The enthalpy, entropy and Gibbs free energy of mixing for the system (SrO +
TiO2) at T = 1200 K is shown as a function of composition in figure 8. The minimum
value of enthalpy and Gibbs free energy of mixing occurs at the equimolar composition corresponding to the compound SrTiO3. It is seen that the compounds
Sr4Ti3O10, Sr3Ti2O7, and Sr2TiO4 are only marginally more stable than mixtures of
SrTiO3 and SrO corresponding to these compositions. This explains why it is relatively easy to synthesize single phase SrTiO3 from component oxides, but rather difficult to prepare pure phases of Sr4Ti3O10, Sr3Ti2O7, and Sr2TiO4. The shape of the
FIGURE 7. The standard Gibbs free energy of formation of Sr2TiO4 from its
component binary oxides: —d— (red online), this study; and —j— (black online),
Knacke et al. [25]. (For interpretation of the references in colour in this figure
legend, the reader is referred to the web version of this article.)
FIGURE 8. Composition dependence of s (blue online), the enthalpy; –h–
(red online), entropy; and —N— (black online), Gibbs free energy of mixing at
1200 K for the system (SrO + TiO2). (For interpretation of the references in colour in
this figure legend, the reader is referred to the web version of this article.)
Neumann–Koop rule to estimate the heat capacity of Sr3Ti2O7 and combining with
data for SrO and TiO2 [23], the standard enthalpy of formation of Sr3Ti2O7 from elements at T = 298.15 K is (3951.441 ± 10.239) kJ mol1. There has been no previous
measurement of either the Gibbs free energy or enthalpy of formation of Sr3Ti2O7.
From the ‘‘second-law” entropy change for reaction (14), the standard entropy of
Sr3Ti2O7 at T = 298.15 K can be evaluated as (257.122 ± 1.268) J mol1 K1.
The EMF of cell IV is related to the standard Gibbs free energy change for the
reaction
SrO ðsolidÞ þ Sr3 Ti2 O7 ðsolidÞ ! 2Sr2 TiO4 ðsolidÞ;
ð16Þ
1
DGrð16Þ 90 ðJ mol Þ ¼ 2FEIV ¼ 32943 2:482ðT=KÞ:
ð17Þ
The standard Gibbs free energy of formation of Sr2TiO4 from its component binary
oxides can be obtained by combining equations (14) and (16). For the reaction
2SrO ðsolidÞ þ TiO2 ðsolidÞ ! Sr2 TiO4 ðsolidÞ;
ð18Þ
1
DGrð18Þ 110 ðJ mol Þ ¼ 159385 þ 3:77ðT=KÞ:
ð19Þ
TABLE 3
Thermodynamic properties of solid Sr2TiO4.
T/K
C P =ðJ K1 mol
298.16
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
1600
1700
1800
143.684
143.984
155.089
161.092
165.087
168.134
170.675
172.922
174.987
176.933
178.798
180.605
182.371
184.106
185.818
187.512
189.192
1
Þ
(HT H298.15)/(kJ mol1)
S/(J K1 mol1)
DHf =ðkJ mol1 Þ
DGf =ðkJ mol1 Þ
0
0.266
15.288
31.121
47.441
64.108
81.051
98.233
115.630
133.226
151.014
168.984
187.133
205.457
223.954
242.620
261.456
167.518
168.408
211.546
246.853
276.596
302.282
324.903
345.138
363.465
380.235
395.711
410.094
423.543
436.185
448.121
459.437
470.203
2281.195
2281.184
2280.177
2278.857
2277.543
2276.340
2275.274
2275.641
2274.357
2288.967
2293.663
2293.732
2293.762
2293.763
2293.748
2567.561
2563.831
2166.429
2165.718
2127.366
2089.314
2051.531
2013.959
1976.550
1939.117
1901.800
1863.871
1825.075
1786.020
1746.964
1707.908
1668.853
1627.440
1572.243
DHf and DGf are the standard enthalpy and Gibbs free energy of formation of Sr2TiO4 from elements.
K.T. Jacob, G. Rajitha / J. Chem. Thermodynamics 43 (2011) 51–57
enthalpy curve suggests that all the four compounds would be thermodynamically
stable at low temperatures. The entropy of mixing has a mild negative value for the
four compositions. The results of this study when extrapolated to higher temperatures suggests the solid-state decomposition of Sr4Ti3O10 to a mixture of SrTiO3 and
Sr3Ti2O7 at T = 1764 ± 9 K, and the decomposition of Sr3Ti2O7 to a mixture of SrTiO3
and Sr2TiO4 at T = 1925 ± 10 K.
4. Conclusions
Accurate values for the standard Gibbs free energies of formation of solid SrTiO3, Sr4Ti3O10, Sr3Ti2O7, and Sr2TiO4 from component binary oxides were determined in the temperature range
(900 to 1250) K. Used for the measurements were solid-state electrochemical cells incorporating single crystal SrF2 as the electrolyte, gold electrodes and pure SrO as the reference electrode and
operated under pure oxygen at ambient pressure. The reference
state for TiO2 is the rutile form. The free energy of formation of
SrTiO3 obtained in this study is in good agreement with direct
measurement of Taylor and Schmalzried [24], but differs from that
given in the compilation of Knacke et al. [25]. The enthalpy of formation of SrTiO3 obtained in this study agrees with solution calorimetric measurements of Xu et al. [21], but differs significantly
from the bomb calorimetric studies of Panfilov and Feodosev
[22]. The incorrect values of free energy of SrTiO3 in the compilation of Knacke et al. [25] result from the wrong selection of enthalpy of formation based on bomb calorimetry. The entropy of
formation of SrTiO3 obtained in this study is consistent with heat
capacity data available in the literature.
For Sr4Ti3O10 values for the standard Gibbs free energy of formation obtained in this study are substantially more negative than
the value suggested by Taylor and Schmalzried [24], and significantly more positive than values given in the compilation of
Knacke et al. [25]. For Sr3Ti2O7 results of this study provide new
information on Gibbs free energy, enthalpy and entropy of formation, which are not available in the literature. The values of standard Gibbs free energy of formation obtained in this study for
Sr2TiO4 are significantly more negative than those given in the
compilation of Knacke et al. [25]. In summary, data for strontium
titanates given in the compilation of Knacke et al. [25] require major revision. New tabulated data for SrTiO3 are presented. For the
other strontium titanates values of enthalpy of formation from elements and standard entropy at T = 298 K are evaluated based on
the results obtained the present study. For a more accurate
description of thermodynamic properties of Sr4Ti3O10 and Sr3Ti2O7,
experimental data of on both low- and high-temperature heat
capacities are required. For Sr2TiO4, low-temperature heat capacity
measurements can confirm the value of standard entropy derived
in this study. The new data generated in this study would be useful
57
for calculation of phase diagrams of higher order systems containing the binary (SrO + TiO2).
Acknowledgements
K.T. Jacob is grateful to the Indian National Academy of Engineering, New Delhi, for the award of INAE Distinguished Professorship. G. Rajitha thanks the University Grants Commission of India
for the award of Dr. D.S. Kothari Postdoctoral Fellowship.
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JCT 10-144